{
  "abstract": "Dujella, Jak\\v setić and Pe\\v carić recently established in \\cite{DJP}, a chain of power-sum inequalities, built from Hölder's and Cauchy's inequalities and their converse forms, and applied it to the Fibonacci sequence using the identities $\\sum_{i=1}^nF_i^2=F_nF_{n+1}$ and $\\sum_{i=1}^nF_iF_{i+1}=F_{n+1}^2-\\frac{1+(-1)^n}{2}$. We show that this machinery applies uniformly across the one-parameter family of $k$-Fibonacci and $k$-Lucas numbers of Falcón and Plaza, via the generalized identities \\begin{align*} \\sum_{i=1}^nL_{k,i}^2=\\frac{L_{k,n}L_{k,n+1}-2k}{k},\\qquad \\sum_{i=1}^nF_{k,i}^2=\\frac{F_{k,n}F_{k,n+1}}{k}, \\end{align*} and \\begin{align*} \\sum_{i=1}^nL_{k,i}L_{k,i+1}=\\frac{L_{k,n+1}^2}{k}-k+\\bigl((-1)^n-1\\bigr)\\left(\\frac{2}{k}+\\frac{k}{2}\\right), \\end{align*} which recover the classical Fibonacci and Lucas identities at $k=1$. We further exploit the cross-identity $F_{k,i}L_{k,i}=F_{k,2i}$, valid for every $k\\ge1$, to obtain a Cauchy-Schwarz-type inequality linking the two families that has no counterpart in the Fibonacci-only setting. At $k=1$, this specializes to a fully explicit elementary inequality between ordinary Fibonacci and Lucas numbers, alongside the corresponding Hölder and Cauchy-conversion refinements we obtain for ordinary Lucas numbers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33573",
  "author_total": 2,
  "authors": [
    {
      "name": "Herbert Batte"
    },
    {
      "name": "Prosper Kaggwa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ekxprs3eccrqc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33573",
  "fulltext_url": null,
  "id": "ekxprs3eccrqc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33573",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7214774321",
  "pdf_url": "https://arxiv.org/pdf/2609.33573",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ekxprs3eccrqc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33573",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33573"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "$k$-Fibonacci and $k$-Lucas numbers with the Hölder inequality",
  "title_text": "k-Fibonacci and k-Lucas numbers with the Hölder inequality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/ekxprs3eccrqc",
  "venue": "arXiv",
  "versions": []
}